A new proof of the graph removal lemma (1006.1300v2)

Abstract: Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^h) copies of H can be made H-free by removing o(n^2) edges. We give a new proof which avoids Szemer\'edi's regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.

An Examination of Jacob Fox's Proof of the Graph Removal Lemma

Jacob Fox presents a novel approach to proving the graph removal lemma, which circumvents the need for Szemerédi's regularity lemma, offering improved bounds. The graph removal lemma, an extension of the triangle removal lemma, posits that any graph containing a negligible number of copies of a fixed subgraph $H$, can be made $H$-free by the removal of a similarly negligible fraction of the edges. Fox's approach leads to a better bound on the number of edge removals required, answering inquiries posed by Alon and Gowers.

Core Contributions

Fox's main contribution lies in refining the bounds associated with the graph removal lemma. Specifically, he addresses the inefficiencies inherent in bounds derived from Szemerédi's regularity lemma, where the magnitude is typically expressed as a tower of twos with height proportional to $\epsilon^{-5}$. Fox's proof establishes that the required bound in his version of the lemma is a tower of twos with height proportional to $\log \epsilon^{-1}$, representing a significant improvement.

The proof methodology involves a operation on the mean entropy density of graph partitions, utilizing refinements that increase this density, thereby converging towards a partition where the removal of edges ensures graph $H$-freedom. Key lemmas, such as those focusing on the shattering of vertex pairs and Jensen's defect inequality, underpin this process, offering rigorous mathematical scaffolding for the iterative refinement process.

Theoretical and Practical Implications

The theoretical implications of Fox's work are profound. By establishing a more efficient proof of the graph removal lemma, Fox opens doors for refined bounds in related areas such as property testing and hypergraph theory. Property testing, in particular, benefits because the lemma provides foundational support for algorithms that determine the structural properties of graphs in constant time, a concept highly applicable in computer science and discrete mathematics.

In practical terms, the implications extend to graph algorithms used in data analysis and computer networking, where understanding and managing graph sub-structures is crucial. Improved bounds mean more efficient algorithms, which could lead to advancements in large-scale graph processing applications.

Fox's work also prompts further exploration into removal lemmas in directed and multicolored graphs, as well as their arithmetic counterparts. The parallels drawn between these areas suggest potential advancements in generalized graph theory, pursuing similar efficiencies through alternative proof strategies.

Speculation on Future Developments

Future developments stemming from Fox's work might explore even tighter bounds on removal lemmas, potentially eschewing regularity-based approaches altogether for a range of combinatorial problems. This could lead to novel insights into the interplay between graph partitions and density measures, and their application to diverse fields such as cryptography, network theory, and large database management.

Further research could refine techniques of graph partition refinement, or explore their application to more complex graphs such as hypergraphs or those representing multidimensional datasets. Exploring nonbipartite graphs for superpolynomial lower bounds remains another potential avenue for investigation.

In summary, Jacob Fox's paper elucidates a significant improvement in proving the graph removal lemma without reliance on Szemerédi's regularity lemma, presenting both theoretical elegance and practical utility that continue to evolve in graph theory and related disciplines.